If I'm not mistaken, on Nightmare difficulty the game actually gives greater benefits to the AI - allowing them to essentially cheat - to make the game more difficult to human players. I only play on Nightmare, and find that it merely draws the game out longer rather than making it more difficult if you play correctly.

I believe they receive more gold, experience and health regeneration - I don't have any hard cold facts, just something I've observered during my time with the game.

Yes Hard and Nightmare AI's cheat. They certainly get bonus gold and experience. I dont know about health regen. This is because it is incredibly difficult to make a "good" AI for this game that wont "rape" all human players. So they have 2 AI's. Both are pretty much exactly the same. The only difference being easy AI rarely use teleports so they are easier to kill without stuns.

They could have programmed a better AI but can you imagineplaying against computer oponents who make all the right chopices and do all the right things with all the right timing? It would be brutal.

So as far as Ive noticed this is the difference:

Easy: Basic AI, no/very little teleports.

Normal: Same Basic AI, just uses teleports more so they dont give up kills as easily.

Hard: Same as normal with 1.5x gold and XP

Nightmare: Same as normal but with 2x gold and xp.

As far as I can see nightmare is only playable if you can really rake in the kills. If you are playing a defensive or support demigod or one who does not focus on killing other DG's, nightmare is VERY difficult. The computer outlevels and out equips you VERY quickly, unless they spend a lot of time dead.

Hardshould be all you need in order to get your practice in. The AI gets enough extra gold to pay for as many teleport scrolls as they need and if you are playing well they wont outlevel you. If you are being outleveled on hard, you are most likely doing something wrong.